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Common models for random graphs, such as Erd\H{o}s-R\'{e}nyi and Kroneckergraphs, correspond to generating random adjacency matrices where each entry isnon-zero based on a large matrix of probabilities. Generating an instance of arandom graph based on these models is easy, although inefficient, by flippingbiased coins (i.e. sampling binomial random variables) for each possible edge.This process is inefficient because most large graph models correspond tosparse graphs where the vast majority of coin flips will result in no edges. Wedescribe some not-entirely-well-known, but not-entirely-unknown, techniquesthat will enable us to sample a graph by finding only the coin flips that willproduce edges. Our analogies for these procedures are ball-dropping, which iseasier to implement, but may need extra work due to duplicate edges, andgrass-hopping, which results in no duplicated work or extra edges.  Grass-hopping does this using geometric random variables. In order to usethis idea on complex probability matrices such as those in Kronecker graphs, wedecompose the problem into three steps, each of which are independently usefulcomputational primitives: (i) enumerating non-decreasing sequences, (ii)unranking multiset permutations, and (iii) decoding and encoding z-curve andMorton codes and permutations. The third step is the result of a new connectionbetween repeated Kronecker product operations and Morton codes. Throughout, wedraw connections to ideas underlying applied math and computer scienceincluding coupon collector problems.

Coin-flipping, ball-dropping, and grass-hopping for generating random graphs from matrices of edge probabilities